Question

A tank is in the shape of an inverted cone, having height \(h=2.5 \mathrm{~m}\) and base radius \(r=0.75 \mathrm{~m} .\) If water is poured into the tank at a rate of \(15 \mathrm{~L} / \mathrm{s}\), how long will it take to fill the tank?

Short answer

Answer: The tank will take approximately 117.87 seconds to fill at a rate of 15 L/s.

Step-by-step solution

Find the volume of the cone

To find the volume (V) of the cone, we will use the formula: \(V = \dfrac{1}{3} \pi r^2 h\) Where \(r\) is the base radius, and \(h\) is the height of the cone.

Substitute values and calculate the tank volume

Given \(r = 0.75 \mathrm{~m}\) and \(h = 2.5 \mathrm{~m}\), substitute the values into the volume formula: \(V = \dfrac{1}{3} \pi (0.75)^2 (2.5)\) Now, calculate the volume: \(V ≈ 1.768 \mathrm{~m^3}\)

Convert the volume to liters

1 cubic meter (\(\mathrm{m^3}\)) is equal to 1000 liters. So to convert \(1.768 \mathrm{~m^3}\) to liters, multiply by 1000: \(1.768 \times 1000 ≈ 1768 \mathrm{~L}\)

Find the time to fill the tank

Given the rate of water being poured is \(15 \mathrm{~L/s}\), we can find the time (t) it takes to fill the tank by dividing the total volume by the rate: \(t = \dfrac{\text{total volume}}{\text{rate}}\) Substitute the values and calculate the time: \(t = \dfrac{1768}{15} ≈ 117.87 \mathrm{~s}\)

Interpret the result

The tank will take approximately \(117.87\) seconds to fill at a rate of \(15 \mathrm{~L/s}\).

Key concepts

Volume of a Cone

Calculating the volume of a cone is a fundamental concept in geometry and calculus. The formula to find the volume of a cone is \( V = \frac{1}{3} \pi r^2 h \). This formula accounts for the base area \( \pi r^2 \) multiplied by the height \( h \) and then multiplied by \( \frac{1}{3} \) to adjust for the cone's shape. In simpler terms, this calculation is similar to finding the volume of a cylinder (base area multiplied by height), reduced to a third because of the cone's tapering.To break it down:

• \( r \) is the radius of the cone's base.
• \( h \) is the cone's height.
• \( \pi \) is a constant approximately equal to 3.14159.

For our problem, if you substitute \( r = 0.75 \text{ m} \) and \( h = 2.5 \text{ m} \), you can compute that the tank's volume is approximately \( 1.768 \text{ m}^3 \). This volume represents the full capacity of the tank in its given conical shape.

Unit Conversion

In many mathematical problems, converting between units is essential to achieve an accurate answer that's easy to interpret. Here, we need to convert the volume of the tank from cubic meters (\( \text{m}^3 \)) to liters (\( L \)) since the water flow rate is given in liters per second. Understanding the conversion:

• 1 cubic meter is equivalent to 1000 liters.
• This conversion is critical since mixing units can lead to incorrect conclusions.

To convert \(1.768 \text{ m}^3\) to liters, simply multiply by 1000:\[1.768 \times 1000 = 1768 \text{ L}\]Thus, the tank's capacity is 1768 liters. Knowing how to switch between units will help in various applications beyond just this problem, including science and engineering sectors.

Rate of Change

Understanding the rate of change is crucial when dealing with continuous processes, like filling a tank with water. Here, the rate of change is represented by the rate at which water is being poured into the tank, which is given as \( 15 \text{ L/s} \). To determine how long it takes to fill the tank, calculate the time \( t \) using the formula:\[t = \frac{\text{total volume}}{\text{rate}}\]When you substitute the values from our problem:\[t = \frac{1768 \text{ L}}{15 \text{ L/s}} \approx 117.87 \text{ seconds}\]Analyzing rate of change helps in anticipating the duration or speed of a process, which is vital in fields such as physics, engineering, and economics. Here, this understanding lets us know that under a continuous fill rate of 15 liters per second, the tank will reach its capacity in about 118 seconds.